Shortly after Heritage announced it would score lawmakers’ votes on the National Defense Authorization Act if it includes language to allow certain illegal immigrants to gain citizenship through military service, the Madison Project also weighed in.

The group, which backs conservative candidates for office and fights for conservative causes on Capitol Hill, warned that Rep. Jeff Denham, R-Calif., plans to offer the legislation, known as the ENLIST Act, as an amendment to the NDAA that’s due on the floor in the coming days.

If Denham stands by the pledge to force a floor vote on the issue, the Madison Group is urging lawmakers to not only vote against the underlying bill — should the amendment win adoption — but pre-empt consideration of that amendment by voting against the rule, the procedural measure that allows the NDAA to come to the floor in the first place.

“We must make it clear to Members of Congress that if amnesty is slipped into the NDAA, especially at a time when [President Barack] Obama is nullifying the rule of law, they must oppose the procedural rule to consider the bill,” Daniel Horowitz, the Madison Project’s policy director, wrote in a blog post on Wednesday evening. “That is the only way to defeat this travesty.

“Democrats typically oppose the procedural motion to call up a bill, even if they intend to vote for final passage,” Horowitz continued. “In this case, most of them will support the bill along with many Republicans. Voting against the procedural rule will ensure that the bill does not make it to the floor.”

Horowitz also called the fight over the ENLIST Act one “for the heart and soul of the Republican Party — one that will determine whether we have a viable alternative to an agenda that disrespects our borders, sovereignty, rule of law, and the American taxpayer.”

The mounting pressure from outside groups to keep the ENLIST Act off the floor or risk sinking the NDAA entirely is one that GOP leaders will have to consider going forward.

The NDAA is typically brought up for consideration under an “open rule,” which means that any lawmaker can come to the floor and force a vote on an amendment, assuming it’s ruled germane.

Denham, who supports comprehensive immigration overhaul, brought up the ENLIST Act last year, but agreed to withdraw it from consideration after a jurisdictional scuffle with Judiciary Chairman Robert W. Goodlatte, R-Va., and also in deference to colleagues made uncomfortable by the politically-loaded amendment.

But Denham has stated he won’t be so accommodating this time around, pledging to make members go on record on the issue.

That vow came after the political criticism that erupted over rumors that House Armed Services Chairman Howard “Buck” McKeon, R-Calif., would allow the language of the ENLIST Act to be included in the base bill of the NDAA before it went before the full panel to be marked up. McKeon ultimately declined to do so, which Denham said left him few options besides taking to the floor.